Finding vector-space bases at random

On the existence of weak bases for vector spaces

The set of all pairs of natural numbers is considered as a directed set under the direction: [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. This directed set is used to study convergence of a double series in a sense of Pringsheim and to introduce double bases in topological vector spaces. An introductory study on double bases is presented.

In linear network coding the information is encoded in terms of a basis of a vector space and it is received as a basis of a possible altered vector space. In the constant dimension case Koetter and Kschischang introduced a metric on the Grassmannian and proved efficient and correct decoding in terms of this metric. Here we introduce a second order invariant of the code: the minimum dimension of the linear span of 3 different linear subspaces belonging to the code. This is the case \(s=3\) of a family \(d_s\) and \(d'_s\), \(s\ge 3\), of invariants of network codes. We study these invariants in a case recently proposed by Hansen (the set of all osculating spaces of a Veronese embedding of a finite projective space) and for a related case (the set of osculating spaces to curves of positive genus) with a complete description of the case of elliptic curves and the ones related to the Hermitian curve.

Abstract. In this paper, we give a consistent presentation of dimension properties and properties of bases for vector spaces over distributive lattices. The bases consisting of join irreducible vectors are studied and their uniqueness is proved. Criteria for the following are proved: the set of join irreducible vectors is a generating set for a vector space; this set is a vector space basis; all bases contain the same number of vectors. A criterion of uniqueness for the basis is proved. The basis containing the greatest number of vectors is found. We give a description for all standard bases of a vector space. We prove a theorem allowing one to calculate the space dimension and to find the basis of the smallest number of vectors by known algorithms. These results are applied to vector spaces over chains: we prove that there exists a standard basis, that the basis of join irreducible vectors is the standard basis, that a standard basis is unique. We calculate the dimension of the arithmetic space and describe all bases containing the smallest number of vectors. It is proved that all such bases are standard.

Accessible proof of standard monomial basis for coordinatization of Schubert sets of flags

The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of "standard monomials" for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. Lakshmibai, Musili, and Seshadri gave a standard monomial basis for the smaller vector space generated by the projective coordinates restricted to a Schubert variety. Reiner and Shimozono made this theory more explicit by giving a straightening algorithm for the products of the minors in terms of the right key of a Young tableau. Since then, Willis introduced scanning tableaux as a more direct way to obtain right keys. This paper uses scanning tableaux to give more-direct proofs of the spanning and the linear independence of the standard monomials. In the appendix it is noted that this basis is a weight basis for the dual of a Demazure module for a Borel subgroup of GL(n). This paper contains a complete proof that the characters of these modules (the key polynomials) can be expressed as the sums of the weights for the tableaux used to index the standard monomial bases.

Polynomial zigzag matrices, dual minimal bases, and the realization of completely singular polynomials

Abstract. We investigate the Hopf algebra conjugation, χ, of the mod2 Steenrod algebra, A 2 , in terms of the Hopf algebra conjugation, χ ′ , ofthe mod 2 Leibniz–Hopf algebra. We also investigate the fixed points ofA 2 under χ and their relationship to the invariants under χ ′ . 1. IntroductionThe mod 2 Steenrod algebra A 2 is the free associative graded algebra gen-erated by the Steenrodsquares Sq n [18] of degree n, n ≥ 1, over F 2 subject tothe AdemrelationsSq a Sq b = ⌊ X a/2⌋s=0 b−1−sa−2sSq a+b−s Sq s for 0 < a < 2b.Conventionally, Sq 0 = 1, the multiplicative identity. Topologically, A 2 is thealgebra of stable cohomology operations for ordinary cohomology H ∗ over F 2 .A monomial in A 2 can be written in the form Sq j 1 Sq j 2 ···Sq j r , which weshall denote by Sq j 1 ,j 2 ,...,j r . Admissible monomials form a vector space ba-sis “admissible basis” for A 2 . Milnor [16] determined the graded connectedHopf algebra structure of A 2 by a cocomutative coproduct given by ∆(Sq

The Vector Space Basis Change (VSBC) is an algebraic operator responsible for change of basis and it is parameterized by a transition matrix. If we change the vector space basis, then each vector com- ponent changes depending on this matrix. The strategy of VSBC has been shown to be effective in separating relevant documents and irrelevant ones. Recently, using this strategy, some feedback algorithms have been de- veloped. To build a transition matrix some optimization methods have been used. In this paper, we propose to use a simple, convenient and direct method to build a transition matrix. Based on this method we develop a relevance feedback algorithm. Experimental results on a TREC collection show that our proposed method is effective and generally superior to known VSBC-based models. We also show that our proposed method gives a statistically significant improvement over these models.

In this review I present some recent results on the convergence properties of formal star products. Based on a general construction of a Fréchet topology for an algebra with countable vector space basis I discuss several examples from deformation quantization: the Wick star product on the flat phase space m2n gives a first example of a Fréchet algebraic framework for the canonical commutation relations. More interesting, the star product on the Poincare disk can be treated along the same lines, leading to a non-trivial example of a convergent star product on a curved Kahler manifold.

One kind of generalized measures called quantum measures on finite effect algebras, which fulfil the grade-2 additive sum rule, is considered. One basis of vector space of quantum measures on a finite effect algebra with the Riesz decomposition property (RDP for short) is given. It is proved that any diagonally positive symmetric signed measure $$\lambda $$ on the tensor product $$E\otimes E$$ can determine a quantum measure $$\mu $$ on a finite effect algebra $$E$$ with the RDP such that $$\mu (x)=\lambda (x\otimes x)$$ for any $$x\in E$$ . Furthermore, some conditions for a grade-2 additive measure $$\mu $$ on a finite effect algebra $$E$$ are provided to guarantee that there exists a unique diagonally positive symmetric signed measure $$\lambda $$ on $$E\otimes E$$ such that $$\mu (x)=\lambda (x\otimes x)$$ for any $$x\in E$$ . At last, it is showed that any grade- $$t$$ quantum measure on a finite effect algebra $$E$$ with the RDP is essentially established by the values on a subset of $$E$$ .

We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC POVMs contains weak SIC POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC POVM.

In this paper we present a hybrid (i.e., quantum-classical) adaptive protocol for the storage and retrieval of discrete-valued information. The purpose of this paper is to introduce a procedure that exhibits how to store and retrieve unanticipated information values by using a quantum property, that of using different vector space bases for preparation and measurement of quantum states. This simple idea leads to an interesting old wish in Artificial Intelligence: the development of computer systems that can incorporate new knowledge on a real-time basis just by hardware manipulation.

We present a detailed description of N=2 stationary BPS multicenter black hole solutions for quadratic prepotentials with an arbitrary number of centers and scalar fields making a systematic use of the algebraic properties of the matrix of second derivatives of the prepotential, $\mathcal{S}$, which in this case is a scalar-independent matrix. In particular we obtain bounds on the physical parameter of the multicenter solution such as horizon areas and ADM mass. We discuss the possibility and convenience of setting up a basis of the symplectic vector space built from charge eigenvectors of the $\ssigma$, the set of vectors $(\Ppm q_a)$ with $\Ppm$ $\ssigma$-eigenspace proyectors. The anti-involution matrix $\mathcal{S}$ can be understood as a Freudenthal duality $\tilde{x}=\ssigma x$. We show that this duality can be generalized to "Freudenthal transformations" $$x\to \lambda\exp(\theta \ssigma) x= a x+b\tilde{x}$$ under which the horizon area, ADM mass and intercenter distances scale up leaving constant the fix point scalars. In the special case $\lambda=1$, "$\ssigma$-rotations", the transformations leave invariant the solution. The standard Freudental duality can be written as $\tilde x= \exp(\pi/2 \ssigma) x .$ We argue that these generalized transformations leave also invariant the general stringy extremal quartic form $\Delta_4$, $\Delta_4(x)= \Delta_4(\cos\theta x+\sin\theta\tilde{x})$.

A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose Grothendieck ring contains an integral form of the Heisenberg algebra in infinitely many variables. We construct bases of vector spaces of morphisms between products of generating objects in this category.

Let (V) be a vector space with a well-ordered basis and $\mathfrak{J}$ be a family of subspaces of (V) closed under intersections. An analog of the Groebner basis is defined for subspaces from $\mathfrak{J}$ . It is shown that in the Noetherian case, such a basis always exists and is unique.

For let be the polynomial algebra in variables of degree one, over the field of two elements. The mod-2 Steenrod algebra acts on according to well-known rules. Let denote the image of the action of the positively graded part of A major problem is that of determining a basis for the quotient vector space Both and are graded where denotes the set of homogeneous polynomials of degree A spike of degree is a monomial of the form where for each In this paper we show that if and can be expressed in the form with then where is the number of spikes of degree

Explicit flows associated to any tangent vector fields on any isospectral manifold for the AKNS operator acting in L2 × L2 on the unit interval are written down. The manifolds are of infinite dimension (and infinite codimension). The flows are called isospectral and also are Hamiltonian flows. It is proven that they may be explicitly expressed in terms of regularized determinants of infinite matrix-valued functions with entries depending only on the spectral data at the starting point of the flow. The tangent vector fields are decomposed as ∑ξkTk where ξ ∈ ℓ2 and the Tk ∈ L2 × L2 form a particular basis of the tangent vector spaces of the infinite dimensional manifold. The paper here is a continuation of Amour [“Explicit isospectral flows for the AKNS operator on the unit interval,” Inverse Probl. 25, 095008 (2009)]10.1088/0266-5611/25/9/095008 where, except for a finite number, all the components of the sequence ξ are zero in order to obtain an explicit expression for the isospectral flows. The regularized determinants induce counter-terms allowing for the consideration of finite quantities when the sequences ξ run all over ℓ2.

We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus.